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arXiv · 2412.08436

On Poincar\'e polynomials for plane curves with quasi-homogeneous singularities

Piotr Pokora
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Abstract

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of such a polynomial over the rationals under the assumption that our conic-line arrangements are free and admit ordinary quasi-homogeneous singularities. Then we focus on the so-called $d$-arrangements in the plane. In particular, we provide a combinatorial constraint for free $d$-arrangements admitting ordinary quasi-homogeneous singularities.

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Piotr Pokora. 2024-12-11. On Poincar\'e polynomials for plane curves with quasi-homogeneous singularities. https://doi.org/10.1112/blms.70112

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